70 problems

Geometric Bogomolov conjecture. If $X$ has dense small points, then $X$ should be special.

Let $K$ be a finitely generated field, let $A$ be the coefficient ring of a Drinfeld module, and let $\phi:A\rightarrow K\{\tau\}$ be a Drinfeld module. For a point…

Let $A$ be an abelian variety defined over a number field $K$, and let $\mathcal{L}$ be a symmetric ample line bundle on $A$. For a point $P\in A(\overline{K})$, let $D=[K(P):K]$ a…

Vojta's multiplier-ideal conjecture. For every real $\varepsilon>0$ and positive integer $r$, there is a proper Zariski-closed subset $Z\subsetneq\mathcal X$, depending only on…

Let $K$ be a number field and let $c>0$. Fix a type $\Phi=(w,(h^r)_{r\in\mathbb{Z}})$ of motives, with $h^r=\dim \operatorname{gr}_{dR}^r$, and choose $a,b$ such that $h^r=0$ unles…

Lang's conjecture. There is a positive constant $c(k)$, depending only on $k$, such that

Let $(X,f,\mathscr L)$ be a polarized dynamical system defined over a number field, and let $Y$ be a subvariety of $X$. Let the maximal preperiodic subvarieties contained in $Y$ be…

Let $K=\mathbb{C}(B)$ be the function field of a smooth projective curve $B$ over $\mathbb{C}$. Let $X$ be a projective variety over $K$ that does not contain any possibly singular…

Let $X$ be a smooth proper variety over a number field $k$, let $D$ be a normal crossings divisor, and let $A$ be an ample line bundle on $X$. Let $r$ be a positive integer and let…

Bogomolov conjecture. If $X_n$ is Zariski dense in $X$ for every $n\geq 1$, then $X$ is a torsion subvariety of $\mathbb{G}_a^g$. This is presented as a consequence of the proposed…

Let $\phi:A\rightarrow K\{\tau\}$ be a Drinfeld module of generic characteristic, let $g\geq 1$, and let $\{x_k\}\subset\mathbb{G}_a^g(K^{\operatorname{sep}})$ be a strict sequence…

Let $K=\mathbb C(B)$, where $B$ is a smooth projective curve, and let $X$ be a projective variety over $K$ containing no rational curve. Let $h_L$ be a Weil height attached to an a…

Let $K=k(B)$, where $B$ is a smooth projective curve, and let $X$ be a projective variety of general type over $K$. A sequence of $K$-rational points in $X$ is generic if every inf…

Multi-indexed generalized Fermat's conjecture. If there is $I_1\geqslant I_0$ such that $Y_I(K)$ is finite for every $I\geqslant I_1$, then there exists $I_2\geqslant I_0$ such tha…

Generalized Fermat's conjecture. If there is $N_1\geqslant N_0$ such that $Y_N(K)$ is finite for every integer $N\geqslant N_1$, then there exists $N_2\geqslant N_0$ such that…

Let $K$ be a number field, let $\mu$ be an adelic measure defined over $K$, let $h_\mu$ be its associated height on $\mathbb{P}^1(\overline{K})$, and let $\mathcal{L}(\mu)$ denote…

Schinzel's conjecture. There exists a nonzero vector ${\bf b}\in\mathbb{Z}^n$ orthogonal to ${\bf a}$ such that

Let $\mathbb{K}$ be a number field with set of places $\mathfrak{M}$, let $\mathbb{T}$ be a split algebraic torus over $\mathbb{K}$ of dimension $n$ with character lattice $M$, and…

Let $S$ be a variety over a number field $K$, and let $L$ be the biextension line bundle on $S$ whose adelic fiberwise structures have degrees equal to the Beilinson–Bloch heights…

Let $S$ be a quasi-projective variety over a number field $K$, let $f\colon X\to S$ be a smooth projective family with relatively ample line bundle $N$, and let $Z$ be the family o…

Let $S$ be a Shimura variety, let $F/E$ be a finite type extension of the reflex field of $S$, let $s=[x,1]\in S(F)$, and let $\Sigma^g(s)$ be a geometric hybrid orbit. Let $w(x')$…

Let $S$ be a Shimura variety, let $F$ be a field of finite type over the reflex field of $S$, and let $s=[x,1]\in S(F)$. Let $\Sigma^g(x)$ denote the geometric hybrid orbit of $x$,…

Let $\mathbb H^3$ be the upper half-space model of hyperbolic $3$-space, and let $\Gamma$ be a geometrically finite subgroup of…

Fix a positive integer $m$. For an algebraic integer $\alpha\in\overline{\mathbb{Q}}$, define … where $M_{\sigma(K)}(\alpha)$ is the relevant relative Mahler measure; for $m=1$, th…

Let $\mathcal{L}/\mathbb{Q}$ be an infinite extension. For a prime power $q$, let $\psi_q$ denote the limiting normalized number of prime ideals of norm $q$ in finite subextensions…